Abstract
This paper has three goals. The first goal is to work out the difference between literal ceteris paribus (cp) laws in the sense of “all others being equal” and ceteris rectis (cr) “laws” in the sense of “all others being right” (Sects. 2, 4). While cp laws involve a universal quantification, cr generalizations involve an existential quantification over the values of the remainder variables Z. As a result, the two differ crucially in their confirmability and lawlikeness. The second goal is to provide a classification of different kinds of cr generalizations (indefinite, definite and normic), including certain transition cases between cr generalizations and cp laws (Sect. 3). The third goal is to work out what cp laws and all kinds of cr assertions have in common: they figure as an information source for assertions of causal influence between variables (Sect. 5).
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Notes
See Barrow (1979, ch. 1.9 and fig. 1–8).
The equivalence of a definite cr law with an ordinary law cannot be read off from its semantic content, but follows from the background knowledge to which the cr clause refers. So it is still appropriate to distinguish between a definite cr law and an ordinary law.
That the gravitational force between two masses has the same spatial direction as their position difference holds in Newton’s law of gravitation and may fail in relativistic mechanics (s. French 2007, ch. 7.4); but Newton’s 2nd law and hence the monotonic cp relationship still holds.
More precisely, D is the set of possible outcomes of a random experiment, and P(Xa = x) is the limiting frequency of result x in an infinite random sequence of D-elements. As usual in statistical probability theory we define P over an algebra over D, P:Al(D)→[0,1]. In the literature on Bayes nets, P is usually defined over an algebra over the combined value space of given random variables, P: Al(⊗ Val) → [0,1], where ⊗ Val = def Val(X1) × ⋯ ×Val(Xn). The latter distribution function is recoverable from the former by the definition P(V1 ×⋯× Vn) = def P({d ∈ D: Xi(d) ∈ Vi, 1 ≤ i ≤ n}) (with Vi ⊆ Val(Xi)), provided the variables are measurable in the sense that for all V1 × ⋯ ×Vn ∈ Al(⊗ Val), {d ∈ D: Xi(d) ∈ Vi, 1 ≤ i ≤ n} ∈ Al(D). This is what we assume; so our definition coheres with the usual Bayes nets formalism.
“X has a causal influence on Y conditional on Z = z” means that in the underlying causal model Y is a cause of X and DEP(X,Y|z) holds.
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Acknowledgments
For valuable discussion I am grateful to Nancy Cartwright, Alexander Gebharter, Andreas Hüttemann, Bernhard Nickel, Jeff Pelletier, Alexander Reutlinger, Jonah Schupbach, Wolfgang Spohn, Michael Strevens, Paul Thorn and Matthias Unterhuber.
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Appendix
Appendix
Proof of the equivalence of “DEP” and “Cr” stated in (17)
DEP ⇒ Cr: If (for some given x, y, z) P(y|x, z) ≠ P(y|z), then by probability theory there must also exist a y*∈Val(Y) such that P(y*|x,z) > P(y*|z). By letting Δx = {x} and Δz = {z} this implies an cr assertion of the form (13)(i) (with empty C).
Cr ⇒ DEP: Assume “Cr”, i.e. for some Δy, Δx and Δz, P(Y ∈ Δy | X ∈ Δx, Z ∈ Δz) > P(Y ∈ Δy|Z ∈ Δz). Y ∈ Δy is equivalent with an exclusive disjunction Y ∈ y1 \( \dot{ \vee } \) ··· \( \dot{ \vee } \)Y ∈ yn; and likewise for X ∈ Δx and Z ∈ Δz. By probability theory, P(A1 \( \dot{ \vee } \)A2|B1 \( \dot{ \vee } \)B2) > P(A1 \( \dot{ \vee } \)A2) implies that P(A1|B1) > P(A1) or P(A2|B2) > P(A2) or P(A1|B2) > P(A1) or P(A2|B1) > P(A2). By generalizing this result to n-fold disjunctions one derives from “Cr”: ∃x,y,z: P(Y = y|X = x, Z = z) > P(Y = y|Z = z); i.e. DEP(X,Y|Z).□
Proof of theorem (18)
By (17), the cr-claim in (18) is equivalent with “DEP(X,Y|Z)”. If (V,E,P) satisfies the causal Markov condition (cMC), then (V,E,P) satisfies the following principle (D) of “d-connection”:
(D) For all X,Y∈V and Z ⊆ V − {X,Y}: If DEP(X,Y|Z), then X and Y are connected by some path π such that no intermediate cause Z (Z’ → Z→ Z’’) or common cause Z (Z’ ← Z→ Z’’) on π is in Z, and every common effect Z on π (Z’ → Z ← Z’’) is in Z or has an effect in Z.
This is a consequence of the first part of theorem 1.2.4 in Pearl (2009, 18).
For (18.1): By (D), DEP(X,Y|Z) entails that
(*) there exists a path π = X − C1 − ··· − Cn − Y (with n ≥ 0) such that no common or intermediate cause Ci on this path π is in Z, and every common effect on π is in Z.
By our assumption,
(**) Z contains all and only the non-effects of X in V − {X,Y}.
We have two cases:
Case (1): If X − Y holds, then X → Y and Y → X are compatible with (*) and (**).
Case (2): X − C1 − ··· −Cn − Y holds for n ≥ 1.
Step (a): Then for no Ci (1 ≤ i ≤ n), X − ···←Ci −− Y can hold in E, for then Ci would be a non-effect of X and a common or intermediate cause on path π, which by (*) must not be in Z. But this contradicts (**).
Step (b): By step (a), X → C1 →→ Cn − Y must hold.
Step (c): X →→ Cn ← Y cannot hold, since then Cn would be an effect of X and a common effect on π, which by (*) must be in Z, in contradiction of (**). Therefore in case (2), X →→ Y must hold.
For (18.2): This is an obvious consequence of (18.1), since t(X) < t(Y) and the assumption that causal arrows are forward-directed in time precludes the second possibility X ← Y.
For (18.3): Spirtes et al. (2000, 50) define an intervention variable (“policy variable”) IX for X as a direct cause of X which not connected with any other variable in V. So if the second possibility of (18.1), X ← Y, would hold, the causal structure would be IX → X ← Y; but then by (D) and since IX is not connected with any other variable in V, DEP(IX,Y) could not hold.□
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Schurz, G. Ceteris Paribus and Ceteris Rectis Laws: Content and Causal Role. Erkenn 79 (Suppl 10), 1801–1817 (2014). https://doi.org/10.1007/s10670-014-9643-8
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DOI: https://doi.org/10.1007/s10670-014-9643-8